Agree with the above. My D came off two waitlists for MT (one in March and another one in early May!). She declined both having accepted elsewhere.
In music, I would not say that it is “slim to none”. I would guess the list isn’t extensive (still I don’t know). If it is a top choice for him, definitely let admissions know. Continue to be enthusiastic and helpful (need anything else to help you decide?). If/when they start making the waitlist calls, it could move him up the list.
There is a chance a teacher doesn’t know (it is possible). If your S liked the teacher and felt like he had a good rapport, it is fine to send a brief “update” email saying that he just wanted the teacher to know that he is grateful that he was waitlisted…since he is still very interested in attending. Again ask if the teacher needs anything more…with a comment at the end that the teacher is not expected to reply to the email.
It can’t hurt to show sincere enthusiasm. I’m sure it hurts a bit…but he’ll have to get used to the hustle part…so better now than never. If he wants it, he should communicate that clearly, briefly and “professionally”.
Good luck!!